An improved particle swarm optimization with double-bottom chaotic maps for numerical optimization

Abstract

Chaos theory studies the behavior of dynamical systems that are highly sensitive to their initial conditions. This effect is popularly referred to as the butterfly effect. Small differences in the initial conditions yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general. In mathematics, a chaotic map is a map (i.e., an evolution function) that exhibits some sort of chaotic behavior. Chaotic maps occur in the study of dynamical systems and often generate fractals. In this paper, an improved logistic map, namely a double-bottom map, with particle swarm optimization was applied to the test function. Simple PSO adopts a random sequence with a random starting point as a parameter, and relies on this parameter to update the positions and velocities of the particles. However, PSO often leads to premature convergence, especially in complex multi-peak search problems. In recent years, the use of chaotic sequences in optimization techniques rather than random sequences with random seeds has been growing steadily. Chaotic sequences, which are created by means of chaotic maps, have been proven easy and fast to generate and are more easily stored then random seed processes. They can improve the performance of PSO due to their unpredictability. Double-bottom maps are designed by the updating equation of PSO in order to balance the exploration and exploitation capability. We embedded many commonly used chaotic maps as well as our double-bottom map into PSO to improve performance, and compared these versions to each other to demonstrate the effectiveness of the PSO with the double-bottom map. We call this improved PSO method Double-Bottom Map PSO (DBMPSO). In the conducted experiments, PSO, DBMPSO and other chaotic PSOs were extensively compared on 22 benchmark test functions. The experimental results indicate that the performance of DBMPSO is significantly better than the performance of other PSOs tested.